Page 466 - Electromagnetics

Expansion of the trigonometric functions then gives
z
sin k(z − ζ)
˜ ˜
ψ(x, y, z,ω) = S(x, y,ζ,ω) dζ +
0 k
sin kz 0 cos kz 0
+ ˜ S(x, y,ζ,ω) cos kζ dζ − ˜ S(x, y,ζ,ω) sin kζ dζ −
2k −∞ 2k −∞
sin kz ∞ cos kz ∞
˜
− ˜ S(x, y,ζ,ω) cos kζ dζ + S(x, y,ζ,ω) sin kζ dζ.
2k 0 2k 0
The last four integrals are independent of z, so we can represent them with functions
constant in z. Finally, rewriting the trigonometric functions as exponentials we have
z sin k(z − ζ)
˜ ˜ ˜ − jkz ˜ jkz
ψ(x, y, z,ω) = S(x, y,ζ,ω) dζ + A(x, y,ω)e + B(x, y,ω)e .
0 k
(A.25)
˜
This formula for ψ was found as a solution to the inhomogeneous ordinary differential
equation (A.19). Hence, to obtain the complete solution we should add any possible
solutions of the homogeneous differential equation. Since these are exponentials, (A.25)
˜
˜
in fact represents the complete solution, where A and B are considered unknown and
can be found using the boundary conditions.
If we are interested in the frequency-domain solution to the wave equation, then we
are done. However, since our boundary conditions (A.17) and (A.18) pertain to the time
domain, we must temporally inverse transform before we can apply them. Writing the
sine function in (A.25) in terms of exponentials, we can express the time-domain solution
as
˜
˜
z
c S(x, y,ζ,ω) c S(x, y,ζ,ω)
ω
ω
˜ −1 j (z−ζ) − j (z−ζ)
ψ(x, y, z, t) = F e c − e c dζ +
0 2 jω 2 jω
−1 ˜ − j z −1 ˜ j z
ω

ω
+ F A(x, y,ω)e c + F B(x, y,ω)e c . (A.26)
A combination of the Fourier integration and time-shifting theorems gives the general
identity

˜ t−t 0
S(x, y,ζ,ω)
−1 − jωt 0
F e = S(x, y,ζ,τ) dτ, (A.27)
jω
−∞
˜
where we have assumed that S(x, y,ζ, 0) = 0. Using this in (A.26) along with the time-
shifting theorem we obtain
ζ−z z−ζ
z

c t− c t− c
ψ(x, y, z, t) = S(x, y,ζ,τ) dτ − S(x, y,ζ,τ) dτ dζ +
2 0 −∞ −∞
z z

+ a x, y, t − + b x, y, t + ,
c c
or
z−ζ
z
c t+ c z z

ψ(x, y, z, t) = S(x, y,ζ,τ) dτ dζ + a x, y, t − + b x, y, t + (A.28)
2 0 z−ζ c c
t−
c
where
−1 ˜ −1 ˜
a(x, y, t) = F [A(x, y,ω)], b(x, y, t) = F [B(x, y,ω)].
© 2001 by CRC Press LLC